Kenny McLean scored a deflected late equaliser to deny Brentford a third win in four games and a place in the top six.
The Canaries substitute tried his luck from outside the area and his shot took a wicked deflection off Mads Bech Sorensen to leave David Raya helpless.
Ivan Toney had scored his eighth goal of the season, from Bryan Mbeumo’s cross, to give Brentford the lead and he was denied a second by a superb save from Tim Krul, who later blocked Henrik Dalsgaard’s shot.
Grant Hanley missed a great chance for the visitors before half-time, but McLean’s late strike left the Bees frustrated and 11th in the table with 11 points from eight games.
Norwich had the better of the early stages and Brentford struggled to play the ball out of defence cleanly.
However, despite their early dominance the Canaries only had two chances, with David Raya touching Emi Buendia’s shot over the bar, and then Mario Vrancic driving just wide of the post.
Having weathered the storm, Brentford took the lead when Saman Ghoddos found Mbeumo and his right-footed cross from the right-hand side of the area was finished off from close range by Toney.
The goal gave the Bees confidence and suddenly they were on top.
Mathias Jensen shot wide and Emiliano Marcondes headed past before Toney’s on-target downward header was superbly tipped away at full stretch by Krul.
Hanley should have equalised two minutes before half-time when left unmarked at Buendia’s free-kick, but he headed over the bar.
Dalsgaard should have made the game safe in the 63rd minute when Toney headed Mbeumo’s cross to him, but the right-back shot straight at Krul.
The visitors won a series of free-kicks but failed to make the most of them, and it looked as though Brentford would see the game out, before McLean struck in the 87th minute.
Brentford: Raya, Dalsgaard, Henry, Sorensen, Pinnock, Jensen, Dasilva (Janelt 70), Marcondes, Ghoddos (Canos 59), Mbeumo, Toney (Forss 83).